🧩 Prime Factorization, LCM, GCF

What is Prime Factorization?
Prime factorization is breaking down a number into its prime factors – the prime numbers that multiply together to make the original number.
Prime Numbers: Numbers greater than 1 with exactly 2 factors (1 and itself)
• Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

Composite Numbers: Numbers with more than 2 factors
• Examples: 4, 6, 8, 9, 10, 12...

What is a Factor Tree?
A factor tree is a diagram that shows how a number breaks down into prime factors.
Steps to Build a Factor Tree:
• Write the number at the top
• Find two factors that multiply to make the number
• Branch them down
• Continue breaking composite numbers until all "leaves" are prime
• Circle the prime numbers

What is GCF?
The Greatest Common Factor (GCF) is the largest number that divides evenly into two or more numbers.
Method 1: List Factors
Find GCF of 18 and 24:
• Factors of 18: 1, 2, 3, 6, 9, 18
• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Common factors: 1, 2, 3, 6
• Greatest common factor: 6

Method 2: Prime Factorization
Find GCF of 18 and 24:
• 18 = 2 × 3²
• 24 = 2³ × 3
• Take the smallest power of each common prime:
• For 2: min(1, 3) = 2¹
• For 3: min(2, 1) = 3¹
• GCF = 2 × 3 = 6

Method 3: Factor Tree Method
18 24
/ / \
2 9 4 6
/ / / \
3 3 2 2 2 3

18 = 2 × 3²
24 = 2³ × 3
GCF = 2 × 3 = 6

What is LCM?
The Least Common Multiple (LCM) is the smallest positive number that is a multiple of two or more numbers.
Method 1: List Multiples
Find LCM of 4 and 6:
• Multiples of 4: 4, 8, 12, 16, 20, 24...
• Multiples of 6: 6, 12, 18, 24, 30...
• Common multiples: 12, 24, 36...
• Least common multiple: 12

Method 2: Prime Factorization
Find LCM of 4 and 6:
• 4 = 2²
• 6 = 2 × 3
• Take the largest power of each prime:
• For 2: max(2, 1) = 2² = 4
• For 3: max(0, 1) = 3¹ = 3
• LCM = 4 × 3 = 12

Method 3: Using GCF Formula
LCM(a, b) = (a × b) ÷ GCF(a, b)

Example: LCM(4,6) = (4×6) ÷ GCF(4,6)
= 24 ÷ 2 = 12